Answer
$\frac{3}{5}$
Work Step by Step
Given expression is-
$\cos(\tan^{-1}\frac{4}{3})$
Assuming $\tan^{-1} \frac{4}{3}$ = $u$ , we get-
$\tan u$ = $\frac{4}{3}$, [$-\frac{\pi}{2}\leq u\leq \frac{\pi}{2}$]
Therefore $\cos u$ will be positive.
From second Pythagorean identity-
$1 + \tan^{2} u$ = $\sec^{2} u$
i.e. $1 + \tan^{2} u$ = $\frac{1}{\cos^{2} u}$
i.e. $\cos^{2} u$ = $\frac{1}{1 + \tan^{2} u}$
$\cos u$ = + $\sqrt {\frac{1}{1 + \tan^{2} u}}$
= $\sqrt {\frac{1}{1 + (\frac{4}{3})^{2}}}$
= $\sqrt {\frac{1}{1 + \frac{16}{9}}}$
= $\sqrt {\frac{1}{\frac{9+16}{9}}}$
= $\sqrt {\frac{1}{\frac{25}{9}}}$
= $\sqrt {\frac{9}{25}}$ = $\frac{3}{5}$
i.e. $\cos u$ = $\frac{3}{5}$
i.e. $\cos(\tan^{-1}\frac{4}{3})$ = $\frac{3}{5}$
